1 Introduction

You can download the latest PDF version HERE And the latest DOCX version HERE

This section explains the different algorithmic aproaches that can be taken in order to completely cover any given freeform surface with panels, ideally wood or metal, which are of approximately the same width and rectantular (or nearly rectangular) when flat and that achieve a surface paneling that is not only cost-effective but also watertight.

To be continued…

2 Background

There is very little backgrounnd on this topic without entering direclty into Orlando’s topic Ruled Surfaces. Some background that must be included:

1. Burj Khalifa interior panelling (Meredith and Kotronis 2013)
2. Ghery’s architecture in general uses same width metal sheets to cover entire buildings, although I am not shure if that is not Orlando’s subject either…
3. Denis Shelden thesis on constructability of gherys architecture (Shelden 2002)
4. MAYBE?? Include non-optimized builidng examples to demonstrate the method’s usefulness.
5. Looking for other built examples or previous/further research on the subject.

3 Computer programs using this technique

There are no programs that develop this technique “out of the box”, but it it based on simple algorithms and can be easily reproduced in any of the latest 3D modeling programs that allow any form of scripting (visual or otherwise) to generate and manipulate 3D geometries. Some examples of this might be:

1. Rhino + Grasshopper
2. Revit + Dynamo
3. Houdini
4. 3DMax

There also exist some powerful geometry processing libraries that can help with the task of computing geodesic curves, distances & fields (which are widely used in this chapter) and other libraries oriented to general scientific and mathematical computing, which are usefull when numerical optimization is needed during the process. Some of those libraries are:

1. LibiGL(C++ with Python bindings)
2. CGal (C++)
3. OpenMesh (C++ with Python bindings)
4. NumPy (Python Computing Library)
5. SciPy (Python Scientific Computing Library )

4 Geodesic curves

In differential geometry, a geodesic curve is the generalization of a straight line into curved spaces (see fig. 1).

Also, in the presence of an affine connection, a geodesic is defined to be a curve whose tangent vectors remain parallel if they are transported along it. We will explore the notion of vector parallel transport in the following sections.

For triangle meshes, shortest polylines cross edges at equal angles.

Finding the truly shortest geodesic paths requires the computation of distance fields (see Carmo 2016; Kimmel and Sethian 1998)

4.1 Algorithmic ways of generating geodesics

The computation of geodesics on smooth surfaces is aclassical topic, and can be reduced to two different solutions, depending on the initial conditions of the problem, you can basically find two tipes of problems (Deng 2011):

Initial value problem

given a point \(p \in S\) and a vector \(v \in T_pS\), find a geodesic which is incident with \(p\), such that the tangent vector fo \(g\) at \(p\) is \(v\).

Boundary value problem

given two points \(p_1, p_2 \in S\) find a geodesic \(g\) connecting \(p_1\) and \(p_2\).

Both problems have different ways of being solved either numerically, graphically or by the means of simulations. The initial value problem can be solved using the concept of straightest geodesics (Polthier and Schmies 1998), whereas the boundary value problem has a very close relation with the computation of shortest paths between two points on a surface.

It is important to note that, during this chapter, all surfaces are discretized as triangular meshes (V,E,F) of sufficient precision.
What would that precision be?? %?? distance to reference surf??

5 Developable surfaces

This is very well explained in p.170 of Denis Shelden thesis (Gerard’s suggestion). Explanation is inspired by that section.

It is also important to introduce the concept of the developable surface, a special kind of surface that have substantial and variable normal curvature while guaranteeing zero gaussian curvature, and as such, this surfaces can be unrolled into a flat plane with no deformation of the surface. This surfaces have been an extremely important design element in the practice of known architect Frank Ghery and explained in (Shelden 2002).

6 Geodesic patterns

What are geodesic patterns?

• Patterns made of panels (wood or metal).
• Bent by their weak axis.
• Mounted on a free-form surface.
• Rectangular or cuasi-rectangular when layed flat.
• Water-tight.
• Overall shape is achieved by pure bending. See Fig. 2

6.1 Properties to aim for in panels

6.1.1 Geodesic property

• Long Thin panels that bend about their weak axis
• Zero geodesic curvature
• Represent the shortest path between two points on a surface

6.1.2 Constant width property

• Panels whose original, unfolded shape is a rectangle.
• The only way this can happen is if the entire surface is developable.
• For all other surfaces:
  • Assuming no gaps between panels
  • Panels will not be exactly rectangular when unfolded
  • Requirement: Geodesic curves that guide the panels must have approximately constant distance from thier neighbourhood curves.

6.1.3 Developable (or ‘pure-bending’) property

• Bending panels on surfaces changes the distances in points only by a small amount so,
• A certain amount of twiting is also present in this aplications.
  Some methods in this chapter do not take into account this property.

6.2 Problem Statement

Problem 1

Look for a system of geodesic curves that covers a freeform surface in a way that:

1. They have approximate constant distance with it’s neighbours.
2. This curves will serve as guiding curves for the panels.
3. The panels are to cover the surface with no overlap and only small gaps

Problem 2

Look for a system of geodesic curves in a freeform surface which:

1. Serve as the boundaries of wooden panels.
2. The panel’s deveopment is nearly straight.
3. Those panels cover the surface with no gaps

7 Design strategies for geodesic systems

7.1 Design by parallel transport

This method, described in (Pottmann et al. 2010), allows for the generation of a system of geodesic curves where either the maximum distance or the minimum distance between adjacent points ocurrs at a prescribed location.

In differential geometry, the concept of parallel transport (see fig. 3) of a vector V along a curve S contained in a surface means moving that vector along S such that:

1. It remains tangent to the surface
2. It changes as little as possible in direction
3. It is a known fact that the length of the vector remains unchanged

7.1.1 Procedure

Following this procedure, extremal distances between adjacent geodesics occur near the chosen curve. Meaning:

1. For surfaces of positive curvature, the parallel transport method will yield a 1-geodesic pattern on which the maximum distance between curves will be \(W\).
2. On the other hand, for surfaces of negative curvature, the method will yield a 1-geodesic pattern with \(W\) being the closest (or minimum) distance between them.

The placement of the first geodesic curve and the selection of the initial vector are not trivial tasks. For surfaces with high variations of surfaces, the results might be unpredictable and, as such, this method is only suitable for surfaces with nearly constant curvature. Other solutions might involve cutting the surface into patches of nearly-constant curvature, and applying the parallel transport method independently on each patch.

7.2 Design by evolution & segmentation

Two main concepts are covered in this section, both proposed by (Pottmann et al. 2010): the first, what is called the evolution method, and a second method based on piecewise-geodesic vector fields.

7.2.1 The evolution method

As depicted in: Starting from a source geodesic somewhere in the surface:

• Evolve a pattern of geodesics iteratively computing ‘next’ geodesics.

• ‘Next’ geodesics must fullfil the condition of being at approximately constant distance from its predecesor.

• If the deviation from its predecesor is too great, breakpoints are introduced and continued as a ‘piecewise geodesic’.

• ‘Next geodesics’ are computed using Jacobi Fields

7.2.2 Distances between geodesics

1. No straight forward solution.
   1. Only for rotational surfaces (surfaces with evenly distriuted meridian curves).
2. But a first-order aproximation of this distance can be approximated:

Starting at time \(t=0\) with a geodesic curve \(g(s)\), parametrized by arc-length \(s\), and let it move within the surface.
A snapshot at time \(t=\varepsilon\) yields a geodesic \(g^+\) near \(g\).

\[g^+(s)=g(s)+\varepsilon\mathbf{v}(s)+\varepsilon^2(\ldots)\qquad(1)\]

The derivative vector field \(\mathbf v\) is called a Jacobi field. We may asume it is orthogonal to \(g(s)\) and it is expressed in terms of the geodesic tangent vector \(g'\) as:

\[\mathbf{v}(s)=w(s)\cdot{R_{\pi/2}(g'(s))},\quad\text{where}\;w''+Kw=0\qquad(2)\]

Since the distance between infinitesimally close geodesics are governed by Eq. 2, that equation also goberns the width of a strip bounded by two geodesics at a small finite distance.

Using this principle, you can develop strips whose width \(w(s)\) fulfills the Jacobi equation \(w(s)=\alpha\cosh(s\sqrt{|K|})\)¹ for some value \(K<0\).
Gluing them together will result in a surface of approximate Gaussian curvature.

7.2.3 Obtaining the ‘next’ geodesic

Input:

A freeform surface \(S\), a desired width \(W\) and a starting geodesic curve \(g_0\)

Output:

The ‘next’ computed geodesic on \(S\)

Pseudocode:

Given a valid \(S\), \(W\) and \(g_0\):

1. Sample the curve \(g_0\) at uniformly distritbuted arc-length parameters \(x_i\).
2. Compute a a set of geodesics \(h_i \perp g_0\) startingt points \(p(x_i)\) on \(g_0\).
3. Select a width function \(\omega(s)\) that is closest to a desired target width function \(W(s)\)(Width function can return a fixed value or a computed value using curvature. It could be further improved accomodate other measures).
4. Select a Jacobi field \(\mathbf v(s)\) using \(\omega(s)\) as stated in Eq. 2.
5. ‘Walk’ the lengh of the vector \(\mathbf v(s)\) on \(h_i\) to obtain \(\chi_i\). These points approximate position of the next geodesic "g^+_i".
6. The previous step can have a very big error margin, therefore we must compute \(g^+_i\):
   1. Select a point \(\chi_i\) and name it \(\chi_0\).
   2. We can compute the next geodesic by minimizing the error between the width function \(\omega(s)\) and the signed distance of the computed \(g^+_i\) to \(g_0\).
7. Return computed geodesic \(g^+_i\)

Any given surface can be completely covered in this maner by recursivly computing the next geodesic using the previous, given some margin of error.

7.3 Piecewise-geodesic vectorfields

8 Panels from curve patterns

In this section, we will discuss several ways to generate panels from a system of 1-geodesic curves.

8.1 Tangent-developable method

The notion of Conjugate tangents on smooth surfaces must be defined:

• Strictly related to the Dupin Indicatrix
• In negatively curved areas, the Dupin Indicatrix is an hyperbola whose asymptotic directions (A1, A2)
• Any parallelogram tangentialy circumscribed to the indicatrix defines two conjugate tangents T and U.
• The asymptotic directions of the dupin indicatrix are the diagonals of any such parallelogram.

Initial algorithm is as follows:

For all geodesics \(s_i\) in a given pattern:

1. Compute the tangent developable surfaces \(\rightarrow\Psi_i\)
2. Trim \(\Psi_i\) along the intersection curves with their respective neighbours.
3. Unfold the trimed \(\Psi_i\), obtaining the panels in flat state.

Unfortunately, this method needs to be refined in order to work in practice because:

1. The rulings of tangent developables may behave in weird ways
2. The intersection of the neighbouring \(\Psi_i\)’s is often ill-defined.

Therefore, the procedure was modified in the following way:

1. Compute the tangent developable surfaces \(\Psi_i\) for all surfaces \(s_i \rightarrow i=\text{even numbers}\)
2. Delete all rulings where the angle enclose with the tangent \(\alpha\) is smaller than a certain threshold (i.e. 20º).
3. Fill the holes in the rulings by interpolation (???)
4. On each ruling:
   1. Determine points \(A_i(x)\) and \(B_i(x)\) which are the closest to geodesics \(s_{i-1}\) and \(s_{i+1}\). This serves for trimming the surface \(\Psi_i\).
5. Optimize globaly the positions of points \(A_i(x)\) and \(B_i(x)\) such that
   1. Trim curves are smooth
   2. \(A_i(x)\) and \(B_i(x)\) are close to geodesics \(s_{i-1}\) and \(s_{i+1}\)
   3. The ruling segments \(A_i(x)B_i(x)\) lies close to the original surface \(\Phi\)

8.2 The Bi-Normal Method

The second method for defining panels, once an appropriate system of geodesics has been found on \(\Phi\), works directly with the geodesic curves.

Assume that a point \(P(t)\) traverses a geodesic \(s\) with unit speed, where \(t\) is the time parameter. For each time \(t\) there is:

• a velocity vector \(T(t)\)
• the normal vector \(N(t)\)
• a third vector \(B(t)\), the binormal vector

This makes \(T.N.B\) a moving orthogonal right-handed frame

The surface \(\Phi\) is represented as a triangle mesh and \(s\) is given as a polyline. For each geodesic, the associated surface is constructed according to Fig. 10. Points \(L(t)\) and \(R(t)\) represent the border of the panel, whose distance from \(P(t)\) is half the panel width.

8.3 Method Comparison

See XXX for more info…

9 Stress and strain in panels

The following section investigates the behaviour of a rectangular strip of elastic material when it is bent to the shape of a ruled surface \(\Psi\) un such way that:

The central line \(m\) of the strip follows the ‘middle geodesic’ \(s\) in \(\Psi\)

This applies to both methods defining panels.fig. 11

9.1 Stress formulas

\[ \rho=1/{\sqrt{K}}, \qquad(3)\] \[ d/2\rho\leq C,\quad{with}\quad{C=\sqrt{\sigma _{max}/E}},\qquad(4)\] \[\varepsilon=\frac{1}{2}(d/2\rho)^2+\cdots\qquad(5)\]

9.1.1 MISSING MORE INFO ON STRESS ANALYISIS**

10 Final analysis cost, quality

All strategies must be compared against cost & quality of the different solutions.

10.1 Frequent measures used in the topic

• Bounding-box diagonal of the panels

10.2 Cost variables

Cost should be defined as:

1. ???
2. ???
3. ???

10.3 Quality variables

Quality should be defined as:

1. ???
2. ???
3. ???

10.4 Variable weighting method

Explanation of the weighting of variables?

11 References

11.1 Must include as bibtex references

• Geodesic Lines Grasshopper implementation
• Non-optimized geodesic planks building
• Non-optimized geodesic planks stairwell
• Discrete Geodesic Nets for Modeling Developable Surfaces
• Video
• Add this paper to bib: Discrete Geodesic Nets

Main refs

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1. Question: What is \(\alpha\) in this formula? Missing image↩